A Two-Layer Model for Steady-Amplitude Gravity Waves and Convection Generated by a Thermal Forcing

A. A. M. Sayed School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada

Search for other papers by A. A. M. Sayed in
Current site
Google Scholar
PubMed
Close
and
L. J. Campbell School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada

Search for other papers by L. J. Campbell in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

A two-dimensional two-layer mathematical model is described representing internal gravity waves and convection generated by a thermal forcing in the lower atmosphere. The model consists of an upper layer with stable stratification, a lower layer with unstable stratification, and a thermal forcing in the form of a nonhomogeneous term in the energy conservation equation. Exact analytical solutions are derived for some simple configurations. Depending on the vertical location and depth of the thermal forcing, the model can be used to represent different configurations in which gravity waves are generated by diabatic heating. When the thermal forcing is centered in the lower layer, convective cells are generated in the lower layer, and gravity waves are forced and propagate upward from the interface between the two layers. When the thermal forcing is centered at the interface, the convection in the lower layer is weaker, and gravity waves are forced by the direct effect of the thermal forcing in the upper layer and the influence of the convective cells below. Steady-amplitude solutions for the vertical profile of the gravity waves and convection are derived and generalized to include cases where there is a spectrum of horizontal wavenumbers or vertical wavenumbers or frequencies present.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Dr. L. J. Campbell, campbell@math.carleton.ca

This article has a companion article which can be found at http://journals.ametsoc.org/doi/abs/10.1175/JAS-D-17-0057.1

Abstract

A two-dimensional two-layer mathematical model is described representing internal gravity waves and convection generated by a thermal forcing in the lower atmosphere. The model consists of an upper layer with stable stratification, a lower layer with unstable stratification, and a thermal forcing in the form of a nonhomogeneous term in the energy conservation equation. Exact analytical solutions are derived for some simple configurations. Depending on the vertical location and depth of the thermal forcing, the model can be used to represent different configurations in which gravity waves are generated by diabatic heating. When the thermal forcing is centered in the lower layer, convective cells are generated in the lower layer, and gravity waves are forced and propagate upward from the interface between the two layers. When the thermal forcing is centered at the interface, the convection in the lower layer is weaker, and gravity waves are forced by the direct effect of the thermal forcing in the upper layer and the influence of the convective cells below. Steady-amplitude solutions for the vertical profile of the gravity waves and convection are derived and generalized to include cases where there is a spectrum of horizontal wavenumbers or vertical wavenumbers or frequencies present.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Dr. L. J. Campbell, campbell@math.carleton.ca

This article has a companion article which can be found at http://journals.ametsoc.org/doi/abs/10.1175/JAS-D-17-0057.1

Save
  • Alexander, M. J., and L. Pfister, 1995: Gravity wave momentum flux in the lower stratosphere over convection. Geophys. Res. Lett., 22, 20292032, https://doi.org/10.1029/95GL01984.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Alexander, M. J., and T. J. Dunkerton, 1999: A spectral parameterization of mean-flow forcing due to breaking gravity waves. J. Atmos. Sci., 56, 41674182, https://doi.org/10.1175/1520-0469(1999)056<4167:ASPOMF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Alexander, M. J., J. R. Holton, and D. R. Durran, 1995: The gravity wave response above deep convection in a squall line simulation. J. Atmos. Sci., 52, 22122226, https://doi.org/10.1175/1520-0469(1995)052<2212:TGWRAD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Alexander, M. J., J. H. Beres, and L. Pfister, 2000: Tropical stratospheric gravity wave activity and relationships to clouds. J. Geophys. Res., 105, 22 29922 309, https://doi.org/10.1029/2000JD900326.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Alexander, S. P., T. Tsuda, Y. Kawatani, and M. Takahashi, 2008: Global distribution of atmospheric waves in the equatorial upper troposphere and lower stratosphere: COSMIC observations of wave mean flow interactions. J. Geophys. Res., 113, D24115, https://doi.org/10.1029/2008JD010039.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Baines, P., 1995: Topographic Effects in Stratified Flows. Cambridge University Press, 482 pp.

  • Batchelor, G. K., 1953: The condition for dynamical similarity of motions of a frictionless perfect-gas atmosphere. Quart. J. Roy. Meteor. Soc., 79, 224235, https://doi.org/10.1002/qj.49707934004.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Beres, J. H., 2004: Gravity wave generation by a three-dimensional thermal forcing. J. Atmos. Sci., 61, 18051815, https://doi.org/10.1175/1520-0469(2004)061<1805:GWGBAT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Beres, J. H., 2005: Estimates of mesospheric gravity wave activity over convection from a global model. Adv. Space Res., 35, 19331939, https://doi.org/10.1016/j.asr.2005.04.087.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Beres, J. H., M. J. Alexander, and J. R. Holton, 2002: Effects of tropospheric wind shear on the spectrum of convectively generated gravity waves. J. Atmos. Sci., 59, 18051824, https://doi.org/10.1175/1520-0469(2002)059<1805:EOTWSO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Beres, J. H., M. J. Alexander, and J. R. Holton, 2004: A method of specifying the gravity wave spectrum above convection based on latent heating properties and background wind. J. Atmos. Sci., 61, 324337, https://doi.org/10.1175/1520-0469(2004)061<0324:AMOSTG>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Beres, J. H., R. R. Garcia, R. B. Boville, and F. Sassi, 2005: Implementation of a gravity wave source spectrum parameterization dependent on the properties of convection in the Whole Atmosphere Community Climate Model (WACCM). J. Geophys. Res., 110, D10108, https://doi.org/10.1029/2004JD005504.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bergman, J. W., and M. L. Salby, 1994: Equatorial wave activity derived from fluctuations in observed convection. J. Atmos. Sci., 51, 37913806, https://doi.org/10.1175/1520-0469(1994)051<3791:EWADFF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Booker, J. R., and F. P. Bretherton, 1967: The critical layer for gravity waves in a shear flow. J. Fluid Mech., 27, 513539, https://doi.org/10.1017/S0022112067000515.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chun, H.-Y., and J.-J. Baik, 1998: Momentum flux by thermally induced internal gravity waves and its approximation for large-scale models. J. Atmos. Sci., 55, 32993310, https://doi.org/10.1175/1520-0469(1998)055<3299:MFBTII>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chun, H.-Y., and J.-J. Baik, 2002: An updated parameterization of convectively forced gravity wave drag for use in large-scale models. J. Atmos. Sci., 59, 10061017, https://doi.org/10.1175/1520-0469(2002)059<1006:AUPOCF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chun, H.-Y., I.-S. Song, J.-J. Baik, and Y.-J. Kim, 2004: Impact of a convectively forced gravity wave drag parameterization in NCAR CCM3. J. Climate, 17, 35303547, https://doi.org/10.1175/1520-0442(2004)017<3530:IOACFG>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Clark, T. L., T. Hauf, and J. P. Kuettner, 1986: Convectively forced internal gravity waves: Results from two-dimensional numerical experiments. Quart. J. Roy. Meteor. Soc., 112, 899925, https://doi.org/10.1002/qj.49711247402.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dutta, G., M. C. Ajay Kumar, P. Vinay Kumar, M. Venkat Ratnam, M. Chandrashekar, Y. Shibagaki, M. Salauddin, and H. A. Basha, 2009: Characteristics of high-frequency gravity waves generated by tropical deep convection: Case studies. J. Geophys. Res., 114, D18109, https://doi.org/10.1029/2008JD011332.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fovell, R., D. Durran, and J. R. Holton, 1992: Numerical simulations of convectively generated stratospheric gravity waves. J. Atmos. Sci., 49, 14271442, https://doi.org/10.1175/1520-0469(1992)049<1427:NSOCGS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fritts, D. C., and M. J. Alexander, 2003: Gravity wave dynamics and effects in the middle atmosphere. Rev. Geophys., 41, 1003, https://doi.org/10.1029/2001RG000106.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Gill, A. E., 1982: Atmosphere-Ocean Dynamics. Academic Press, 662 pp.

  • Goldstein, S., 1931: On the stability of superposed streams of fluids of different densities. Proc. Roy. Soc. London, 132A, 524548, https://doi.org/10.1098/rspa.1931.0116.

    • Search Google Scholar
    • Export Citation
  • Hayashi, Y., 1976: Non-singular resonance of equatorial waves under the radiation condition. J. Atmos. Sci., 33, 183201, https://doi.org/10.1175/1520-0469(1976)033<0183:NSROEW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hines, C. O., 1997: Doppler-spread parameterization of gravity-wave momentum deposition in the middle atmosphere. Part 2: Broad and quasi monochromatic spectra, and implementation. J. Atmos. Sol.-Terr. Phys., 59, 387400, https://doi.org/10.1016/S1364-6826(96)00080-6.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holton, J. R., 2004: An Introduction to Dynamic Meteorology. 4th ed. Elsevier, 535 pp.

  • Holton, J. R., J. Beres, and X. Zhou, 2002: On the vertical scale of gravity waves excited by localized thermal forcing. J. Atmos. Sci., 59, 20192023, https://doi.org/10.1175/1520-0469(2002)059<2019:OTVSOG>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jewtoukoff, V., R. Plougonven, and A. Hertzog, 2013: Gravity waves generated by deep tropical convection: Estimates from balloon observations and mesoscale simulations. J. Geophys. Res. Atmos., 118, 96909707, https://doi.org/10.1002/jgrd.50781.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kovalam, S., R. A. Vincent, and P. Love, 2006: Gravity waves in the equatorial MLT region. J. Atmos. Sol.-Terr. Phys., 68, 266282, https://doi.org/10.1016/j.jastp.2005.05.009.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kuester, M. A., M. J. Alexander, and E. A. Ray, 2008: A model study of gravity waves over Hurricane Humberto (2001). J. Atmos. Sci., 65, 32313246, https://doi.org/10.1175/2008JAS2372.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lane, T. P., and M. W. Moncrieff, 2008: Stratospheric gravity waves generated by multiscale tropical convection. J. Atmos. Sci., 65, 25982614, https://doi.org/10.1175/2007JAS2601.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lane, T. P., M. J. Reeder, and T. L. Clark, 2001: Numerical modeling of gravity wave generation by deep tropical convection. J. Atmos. Sci., 58, 12491274, https://doi.org/10.1175/1520-0469(2001)058<1249:NMOGWG>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lindzen, R. S., 1981: Turbulence and stress owing to gravity wave and tidal breakdown. J. Geophys. Res., 86, 97079714, https://doi.org/10.1029/JC086iC10p09707.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McLandress, C. D., M. J. Alexander, and L. Wu, 2000: Microwave Limb Sounder observations of gravity waves in the stratosphere: A climatology and interpretation. J. Geophys. Res., 105, 11 94711 967, https://doi.org/10.1029/2000JD900097.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ogura, Y., and N. A. Phillips, 1962: Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci., 19, 173179, https://doi.org/10.1175/1520-0469(1962)019<0173:SAODAS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pandya, R. E., and M. J. Alexander, 1999: Linear stratospheric gravity waves above convective thermal forcing. J. Atmos. Sci., 56, 24342446, https://doi.org/10.1175/1520-0469(1999)056<2434:LSGWAC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pfister, L., W. Starr, R. Craig, and M. Loewenstein, 1986: Small-scale motions observed by aircraft in the tropical lower stratosphere: Evidence for mixing and its relationship to large-scale flows. J. Atmos. Sci., 43, 32103225, https://doi.org/10.1175/1520-0469(1986)043<3210:SSMOBA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pfister, L., S. Scott, M. Loewenstein, S. Bowen, and M. Legg, 1993: Mesoscale disturbances in the tropical stratosphere excited by convection: Observations and effects on the stratospheric momentum budget. J. Atmos. Sci., 50, 10581075, https://doi.org/10.1175/1520-0469(1993)050<1058:MDITTS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Piani, C., D. Durran, M. J. Alexander, and J. R. Holton, 2000: A numerical study of three-dimensional gravity waves triggered by deep tropical convection and their role in the dynamics of the QBO. J. Atmos. Sci., 57, 36893702, https://doi.org/10.1175/1520-0469(2000)057<3689:ANSOTD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Salby, M. L., and R. R. Garcia, 1987: Transient response to localized episodic heating in the tropics. Part I: Excitation and short-time near-field behavior. J. Atmos. Sci., 44, 458498, https://doi.org/10.1175/1520-0469(1987)044<0458:TRTLEH>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sang, J., 1991: On formation of convective roll vortices by internal gravity waves: A theoretical study. Meteor. Atmos. Phys., 46, 1528, https://doi.org/10.1007/BF01026620.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sato, K., 1992: Vertical wind disturbances in the afternoon of mid-summer revealed by the MU radar. Geophys. Res. Lett., 19, 19431946, https://doi.org/10.1029/92GL02244.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sayed, A. A. M., 2014: Internal gravity waves and convection generated by a thermal forcing in the atmosphere. Ph.D. thesis, Carleton University, 199 pp.

  • Song, I. S., H. Y. Chun, and T. P. Lane, 2003: Generation mechanisms of convectively forced internal gravity waves and their propagation to the stratosphere. J. Atmos. Sci., 60, 19601980, https://doi.org/10.1175/1520-0469(2003)060<1960:GMOCFI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sutherland, B. R., 2010: Internal Gravity Waves. Cambridge University Press, 377 pp.

    • Crossref
    • Export Citation
  • Taylor, G. I., 1931: Effect of variation in density on the stability of superposed streams of fluid. Proc. Roy. Soc. London, 132A, 499523, https://doi.org/10.1098/rspa.1931.0115.

    • Search Google Scholar
    • Export Citation
  • Taylor, M. J., and Coauthors, 2009: Characteristics of mesospheric gravity waves near the magnetic equator, Brazil, during the SpreadFEx campaign. Ann. Geophys., 27, 461472, https://doi.org/10.5194/angeo-27-461-2009.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Tsuda, T., S. Kato, T. Yokoi, T. Inoue, M. Yamamoto, T. VanZandt, S. Fukao, and T. Sato, 1990: Gravity waves in the mesosphere observed with the middle and upper atmosphere radar. Radio Sci., 25, 10051018, https://doi.org/10.1029/RS025i005p01005.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Venkateswara Rao, N., Y. Shibagaki, and T. Tsuda, 2011: Diurnal variation of short-period (20–120 min) gravity waves in the equatorial mesosphere and lower thermosphere and its relation to deep tropical convection. Ann. Geophys., 29, 623629, https://doi.org/10.5194/angeo-29-623-2011.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Vincent, R. A., and M. J. Alexander, 2000: Gravity waves in the tropical lower stratosphere: An observational study of seasonal and interannual variability. J. Geophys. Res., 105, 17 97117 982, https://doi.org/10.1029/2000JD900196.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wurtele, M. G., R. D. Sharman, and A. Datta, 1996: Atmospheric lee waves. Annu. Rev. Fluid Mech., 28, 429476, https://doi.org/10.1146/annurev.fl.28.010196.002241.

    • Crossref
    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 190 88 9
PDF Downloads 189 57 7